L11n386

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_17,19,18,22 X_11,20,12,21 X_19,10,20,11 X_21,5,22,18 X_2,14,3,13
Gauss code	{1, -11, -5, 3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, 11, -2, -3, 6, -7, 10}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-4q^{-6}+6q^{-5}-8q^{-4}+9q^{-3}-6q^{-2}+2q+7q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-a^{6}z^{-2}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+11z^{2}a^{4}+4a^{4}z^{-2}+10a^{4}-3z^{4}a^{2}-10z^{2}a^{2}-5a^{2}z^{-2}-11a^{2}+2z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{9}z^{5}-3a^{9}z^{3}+a^{9}z+2a^{8}z^{6}-5a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+9a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+3a^{6}z^{8}-10a^{6}z^{6}+14a^{6}z^{4}-9a^{6}z^{2}-a^{6}z^{-2}+5a^{6}+a^{5}z^{9}+2a^{5}z^{7}-17a^{5}z^{5}+35a^{5}z^{3}-22a^{5}z+5a^{5}z^{-1}+5a^{4}z^{8}-20a^{4}z^{6}+39a^{4}z^{4}-34a^{4}z^{2}-4a^{4}z^{-2}+16a^{4}+a^{3}z^{9}-8a^{3}z^{5}+26a^{3}z^{3}-26a^{3}z+9a^{3}z^{-1}+2a^{2}z^{8}-8a^{2}z^{6}+23a^{2}z^{4}-32a^{2}z^{2}-5a^{2}z^{-2}+17a^{2}+az^{7}-az^{5}+3az^{3}-10az+5az^{-1}+3z^{4}-8z^{2}-2z^{-2}+7$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

2

1

-1

6

2

4

-3

4

5

1

-5

5

2

3

-7

3

4

1

-9

3

5

-2

-11

1

3

2

-13

1

3

-2

-15

1

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$